The LGMRES algorithm [R387][R388] is designed to avoid some problems
in the convergence in restarted GMRES, and often converges in fewer
iterations.

Parameters:

A : {sparse matrix, dense matrix, LinearOperator}

The real or complex N-by-N matrix of the linear system.

b : {array, matrix}

Right hand side of the linear system. Has shape (N,) or (N,1).

x0 : {array, matrix}

Starting guess for the solution.

tol : float, optional

Tolerance to achieve. The algorithm terminates when either the relative
or the absolute residual is below tol.

maxiter : int, optional

Maximum number of iterations. Iteration will stop after maxiter
steps even if the specified tolerance has not been achieved.

M : {sparse matrix, dense matrix, LinearOperator}, optional

Preconditioner for A. The preconditioner should approximate the
inverse of A. Effective preconditioning dramatically improves the
rate of convergence, which implies that fewer iterations are needed
to reach a given error tolerance.

callback : function, optional

User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.

inner_m : int, optional

Number of inner GMRES iterations per each outer iteration.

outer_k : int, optional

Number of vectors to carry between inner GMRES iterations.
According to [R387], good values are in the range of 1...3.
However, note that if you want to use the additional vectors to
accelerate solving multiple similar problems, larger values may
be beneficial.

outer_v : list of tuples, optional

List containing tuples (v,Av) of vectors and corresponding
matrix-vector products, used to augment the Krylov subspace, and
carried between inner GMRES iterations. The element Av can
be None if the matrix-vector product should be re-evaluated.
This parameter is modified in-place by lgmres, and can be used
to pass “guess” vectors in and out of the algorithm when solving
similar problems.

store_outer_Av : bool, optional

Whether LGMRES should store also A*v in addition to vectors v
in the outer_v list. Default is True.

The LGMRES algorithm [R387][R388] is designed to avoid the
slowing of convergence in restarted GMRES, due to alternating
residual vectors. Typically, it often outperforms GMRES(m) of
comparable memory requirements by some measure, or at least is not
much worse.

Another advantage in this algorithm is that you can supply it with
‘guess’ vectors in the outer_v argument that augment the Krylov
subspace. If the solution lies close to the span of these vectors,
the algorithm converges faster. This can be useful if several very
similar matrices need to be inverted one after another, such as in
Newton-Krylov iteration where the Jacobian matrix often changes
little in the nonlinear steps.